An A.C. (Alternating Current) series circuit is quite complicated. We will focus our attention on one aspect of the system: the frequency-dependent "back electromotive force (e.m.f.)" that determines the magnitude of the average current.
Let's first review the concept of "root mean-squared" (rms) averages of voltages,
currents and powers (from our study of transformers). Voltages and currents in an A.C.
circuit are sinusoidal, with values that are sometimes positive and sometimes negative and
sometimes zero. Power is sinusoidal also, but is always positive-- it's a sine wave with
its troughs at zero value. Ohm's Law for
A.C. resistor circuits can be written in terms of peak values or average values of V and I:
Inductance is defined as the ability of a helical coil of wire to produce a good
magnetic field (storing energy) when a voltage is applied to it. It's value L
is maximized when the coil has many turns N packed into relatively short length l, with
large area of opening A for the turns. The equation for L is
Now let's turn our attention to a series resistor-inductor-capacitor (RLC) A.C. circuit.
The rms current in the circuit depends on both the input voltage and the resistance, but
it also is a function of the frequency of the A.C.. This is because the capacitor and the
inductor both provide back e.m.f. which limits the current flow. Resistors limit
current by causing charges to lose energy via non-conservative force (friction), but
capacitors and inductors store potential energy that is later recovered (hence only
the resistor dissipates power here). The storage of energy is only possible if the
input voltage source does work against a retarding force: what we are calling back e.m.f..
The capacitor provides back e.m.f. when it is charged: it acts like a second "battery" with
polarity that would push current the opposite way compared with the input voltage. Since
capacitors require time to charge fully (recall RC time constant), the back e.m.f. of the
capacitor is only significant if the voltage input remains in one polarity for a while,
giving the capacitor the chance to charge up and "push back" with its own voltage. Thus,
capacitors "push back" best when A.C. frequency is low. Their current limiting
ability is expressed as capacitive reactance XC in ohms. The equation
for capacitive reactance as a function of frequency is
The inductor provides back e.m.f. when the magnetic flux through its coil is changing
rapidly (say from zero initial value). This according to Faraday's Law. So when current
is just beginning to flow through an inductor containing no magnetic field, that's when
the back e.m.f. of the inductor is greatest. As the field in the inductor builds up, the
back e.m.f. diminishes and the current can reach a high value-- given enough time. So
the back e.m.f. of the inductor is overall unimportant if the voltage input remains in
one polarity for a while, allowing the magnetic field in the inductor to fully form and
stopping its flux change thereby letting the current through for much of the cycle. Thus,
inductors "push back" best when A.C. frequency is high. Their current limiting
ability is expressed as inductive reactance XL in ohms. The equation
for inductive reactance as a function of frequency is
When an A.C. circuit contains just a single inductor or a single capacitor, ohms law is: